Communicated by Irving Kaplansky, August 7, 1959
It is well known [l, Chapter XVI, §9, Application 4] that a finite
group which acts freely2 on a finite dimensional homology ^-sphere
must have periodic cohomology with period k + 1. I will outline here
a proof of a converse result : A ny finite group with periodic cohomology
can act freely on a finite simplicial homotopy sphere. More precisely,
T H E O R E M 1. Let w be a finite group having periodic cohomology of
period q. Let n be the order of 7r. Let d be the greatest common divisor of
n and <j>{n), c/> being Ruler's $-function. Then IT acts freely and simplicially on a finite simplicial homotopy (dq—1)-sphere X of dimension
dq—1. Furthermore y if X is not required to be a finite complex, we can
replace dq — 1 by q—1.
Note that a result of Milnor [3] shows that for some groups 7r,
X cannot be a manifold.
The proof of this theorem can be reduced to pure algebra by using
a remark of Milnor (unpublished) to the effect that any free resolution of Z over TT can be realized geometrically provided this is so in
low dimensions. Thus our main problem is to prove the following
T H E O R E M 2. Let w, q, d be as in Theorem 1. Then w has a periodic
free resolution of period dq. Also, T has a projective resolution of period q.
By a periodic free resolution of period k, I mean a Tate complex
(or complete resolution [ l , Chapter XII, §2]) for TT having an automorphism of degree k. The existence of such a resolution is easily seen
to be equivalent to the existence of an exact sequence
0 - • Z -> Wk-x - •
with all Wi being free over the group ring Zir. I t is this sequence (1)
to which we apply Milnor's construction.
As a first step in constructing such a sequence, we choose a free
Sponsored by the Office of Ordnance Research, U. S. Army under contract
A group is said to act freely on a space if no element of the group other than 1
fixes any point of the space.
0 -> A -> Wt-i -> • • • ~> Wi -* Wo -> Z -> 0.
Define two modules ikf and iV over Zir to be equivalent [2 ] (denoted
M~N) if there are finitely generated projectives P and Q over Zx
such that M+P^N+Q.
By a theorem of Schanuel [2], the equivalence class of A in (2) is uniquely determined. This also follows immediately from the following result which generalizes the theorem of
1. Let C and C be two chain complexes {over any ring)
of the form
0 ~> Ck - > C*-i - > • • • - > Co - > 0
0 - > Ci - > C*'-i - >
• Co' - > 0
ztóft a// d and C{ projective except possibly Ck and Ck . Suppose there
is a chain mapf: C—*C inducing isomorphisms of all homology groups.
Then there is an isomorphism
Ck + Cfc'_i + Ck-2 + Ck-z + • • • « C* + Ck-i + Ck-2 + C&_3 + • • • .
Furthermorey this isomorphism can be chosen so that injecting Ck into
the left hand side and projecting the right hand side onto Ck yields the
mapf: Ck—*Ci.
This result is an algebraic analogue of a geometric theorem of
J. H. C. Whitehead [6, Theorem 6]. It was suggested by a discussion
of simple homotopy type with W. H. Cockcroft.
It is now easy to show that a projective resolution of the form
(1) exists if and only if we have A~Z in the sense explained above.
In fact, the following stronger result holds.
LEMMA 1. Suppose A +P^Z+Q
with P and Q projective, the A being that of (2). Then there is an exact sequence of the form
+ P-*Wk-2
+ Q->Wk-z-*
* • •
all maps from Wk-z on being the same as in (2).
Note that this lemma shows that the low dimensional terms in (1)
can be chosen arbitrarily. This insures that Milnor's construction can
be performed without difficulty.
We now need a criterion for the equivalence A^Z.
LEMMA 2. Let A be any finitely generated module over Zir. Then
A~Z if and only if both of the following conditions are satisfied.
(i) Ê°(T, A) has an element of order n, n being the order of w.
(ii) A r^Z as modules over each sylow subgroup of w.
The proof makes use of a theorem of Rim [4, Proposition 4.9].
This yields the statement about projective resolutions in Theorem 2.
The proof of the statement about free resolutions uses a result of
[5, Theorem 4 ] .
Some of the calculations made in connection with this work lead
easily to a simple group-theoretic interpretation of the period of a
group. Recall that for every prime p, there is a ^-period associated
with any finite group [l, Chapter X I I , Example 11 ]. The period of
the cohomology is the least common multiple of the ^-periods. The
^-period is infinite unless the £-sylow subgroup of w is cyclic or generalized quaternion.
T H E O R E M 3. (a) If the 2-sylow subgroup of w is cylic, the 2-period
is 2. If the 2-sylow subgroup is generalized quaternion, the 2-period is 4.
(b) Suppose p is odd and the p-sylow subgroup of w is cyclic. Let
$p be the group of automorphisms of the p-sylow subgroup induced by
inner automorphisms of IT. Then the p-period is twice the order of $>p.
1. H. Cartan and S. Eilenberg, Homologuai algebra, Princeton, 1956.
2. I. Kaplansky, Homologuai dimension of rings and modules, mimeographed
notes, The University of Chicago, 1959.
3. J. W. Milnor, Groups which act on Sn without fixed points, Amer. J. Math. vol.
79 (1957) pp. 623-630.
4. D. S. Rim, Modules over finite groups, Ann. of Math. vol. 69 (1959) pp. 700712.
5. R. G. Swan, Projective modules over finite groups, Bull. Amer. Math. Soc. vol.
65 (1959) pp. 365-367.
6. J. H. C. Whitehead, Combinatorial homotopy I, Bull. Amer. Math. Soc. vol. 55
(1949) pp. 213-245.